Renormalization Group as a Koopman Operator

TL;DR

This paper reveals a direct link between Koopman operator theory and the renormalization group, enabling the computation of critical exponents from single observables without translational invariance, broadening the applicability of RG methods.

Contribution

It establishes a novel connection between Koopman operators and the renormalization group, allowing for data-driven analysis of critical phenomena in classical spin systems.

Findings

Computed critical exponents $ta$ and $elta$ from single observables.

Broadened the applicability of RG to non-translationally invariant systems.

Proposed a new data-driven approach to identify RG fixed points.

Abstract

Koopman operator theory is shown to be directly related to the renormalization group. This observation allows us, with no assumption of translational invariance, to compute the critical exponents $\eta$ and $\delta$, as well as ratios of critical exponents, of classical spin systems from single observables alone. This broadens the types of problems that the renormalization group framework can be applied to and establish universality classes of. In addition, this connection may allow for a new, data-driven way in which to find the renormalization group fixed point(s), and their relevant and irrelevant directions.